Optimal. Leaf size=98 \[ \frac {A (a+b x)}{i^2 (c+d x) (b c-a d)}+\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{i^2 (c+d x) (b c-a d)}-\frac {B (a+b x)}{i^2 (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{d i^2 (c+d x)}+\frac {b B \log (a+b x)}{d i^2 (b c-a d)}-\frac {b B \log (c+d x)}{d i^2 (b c-a d)}+\frac {B}{d i^2 (c+d x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2525
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(42 c+42 d x)^2} \, dx &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac {B \int \frac {b c-a d}{42 (a+b x) (c+d x)^2} \, dx}{42 d}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1764 d}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}+\frac {(B (b c-a d)) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1764 d}\\ &=\frac {B}{1764 d (c+d x)}+\frac {b B \log (a+b x)}{1764 d (b c-a d)}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{1764 d (c+d x)}-\frac {b B \log (c+d x)}{1764 d (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 104, normalized size = 1.06 \[ \frac {-a A d+B (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )-b B (c+d x) \log (a+b x)+a B d+A b c+b B c \log (c+d x)+b B d x \log (c+d x)-b B c}{d i^2 (c+d x) (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 88, normalized size = 0.90 \[ -\frac {{\left (A - B\right )} b c - {\left (A - B\right )} a d - {\left (B b d x + B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x + {\left (b c^{2} d - a c d^{2}\right )} i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 120, normalized size = 1.22 \[ -{\left (\frac {{\left (b x e + a e\right )} B \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {{\left (b x e + a e\right )} {\left (A - B\right )}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 515, normalized size = 5.26 \[ -\frac {B \,a^{2} d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}+\frac {2 B a b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}-\frac {B \,b^{2} c^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{2}}-\frac {A \,a^{2} d}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}+\frac {2 A a b c}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}-\frac {A \,b^{2} c^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{2}}+\frac {B \,a^{2} d}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}-\frac {2 B a b c}{\left (a d -b c \right )^{2} \left (d x +c \right ) i^{2}}-\frac {B a b \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} i^{2}}+\frac {B \,b^{2} c^{2}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d \,i^{2}}+\frac {B \,b^{2} c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} d \,i^{2}}-\frac {A a b}{\left (a d -b c \right )^{2} i^{2}}+\frac {A \,b^{2} c}{\left (a d -b c \right )^{2} d \,i^{2}}+\frac {B a b}{\left (a d -b c \right )^{2} i^{2}}-\frac {B \,b^{2} c}{\left (a d -b c \right )^{2} d \,i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 134, normalized size = 1.37 \[ -B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac {1}{d^{2} i^{2} x + c d i^{2}} - \frac {b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} + \frac {b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} - \frac {A}{d^{2} i^{2} x + c d i^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 106, normalized size = 1.08 \[ -\frac {A-B}{x\,d^2\,i^2+c\,d\,i^2}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{d^2\,i^2\,\left (x+\frac {c}{d}\right )}+\frac {B\,b\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.52, size = 231, normalized size = 2.36 \[ \frac {B b \log {\left (x + \frac {- \frac {B a^{2} b d^{2}}{a d - b c} + \frac {2 B a b^{2} c d}{a d - b c} + B a b d - \frac {B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {B b \log {\left (x + \frac {\frac {B a^{2} b d^{2}}{a d - b c} - \frac {2 B a b^{2} c d}{a d - b c} + B a b d + \frac {B b^{3} c^{2}}{a d - b c} + B b^{2} c}{2 B b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac {- A + B}{c d i^{2} + d^{2} i^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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